A probability density function (PDF) describes the relative likelihood of observing any given sample value of a random variable. The integral of a PDF over all possible values is 1; the integral of a PDF over a subset of the random variable's range expresses the probability that a drawn sample of a random variable will fall within that range.
PDFs that can be expressed by a closed-form equation are generally well understood, and many applications for such PDFs have been developed. On the other hand, the practical estimation of a PDF for a complex multidimensional random variable, particularly one with an unknown and possibly irregular distribution in each dimension, and/or long, sparsely populated tails, has in large part eluded researchers. In the area of pattern and image recognition, for instance, many researchers have abandoned PDF approaches and concentrated on known solvable alternatives, such as Neural Networks and linear discriminant functions, due to the practical difficulties in applying a PDF approach.